Real, Rational, and Irrational Numbers Revision Notes for Grade 10 NSC Matric Mathematics

Real, Rational, and Irrational Numbers

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Understanding the different types of numbers and how they relate to each other is fundamental to mathematics. The real number system organises all numbers we commonly work with into different categories based on their properties.

The real number system

The real number system creates a logical hierarchy where each type of number builds upon the previous types. This systematic organisation helps us understand the relationships between different categories of numbers.

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Understanding the Hierarchy

The diagram above shows how different number sets fit together. Each inner set is completely contained within the outer sets, creating a hierarchy of number types. This nested structure means that every natural number is also a whole number, every whole number is also an integer, and so on.

Number set definitions

Natural numbers (ℕ)

Natural numbers are the counting numbers we use in everyday life: ${1, 2, 3, 4, 5, ...}$ . These are the numbers you would use to count objects or people.

Whole numbers (ℕ₀)

Whole numbers include all natural numbers plus zero: ${0, 1, 2, 3, 4, 5, ...}$ . The zero makes this set "whole" by including the concept of nothing.

Integers (ℤ)

Integers include all whole numbers and their negative counterparts: ${..., -3, -2, -1, 0, 1, 2, 3, ...}$ . This set extends in both directions from zero.

Real numbers (ℝ)

Real numbers include all numbers that can be represented on a number line. This encompasses both rational and irrational numbers.

Non-real numbers

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Non-Real Numbers Warning

Not all numbers are real numbers. The square root of a negative number is called a non-real or imaginary number. For example, $\sqrt{-1}$ , $\sqrt{-28}$ , and $\sqrt{-5}$ are all non-real numbers and cannot be placed on the standard number line.

Rational numbers

Definition

A rational number $(\mathbb{Q})$ is any number that can be expressed as a fraction:

$\frac{a}{b}$

where $a$ and $b$ are integers and $b \neq 0$ .

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Memory Aid: Rational = Ratio

The word "rational" comes from "ratio", helping you remember that these numbers can be written as a ratio of two integers. This connection makes it easier to identify rational numbers.

Key properties

Examples of rational numbers include:

$\frac{10}{1}$ , $\frac{21}{7}$ , $\frac{-1}{3}$ , $\frac{10}{20}$ , $\frac{-3}{6}$

Notice that all integers are also rational numbers because they can be written with a denominator of 1.

Decimal representations of rational numbers

Rational numbers have specific patterns when written as decimals:

Terminating decimals: These end after a certain number of decimal places. For example, $\frac{4}{10} = 0.4$ .

Recurring decimals: These have a repeating pattern of digits. The fraction $\frac{1}{3} = 0.333...$ or $\frac{2}{11} = 0.181818...$

You can use a dot or bar over the repeated digits to show the pattern: $0.\overline{3}$ or $0.\overline{18}$ .

Irrational numbers

Definition

Irrational numbers $(\mathbb{Q}')$ are numbers that cannot be written as a fraction where both the numerator and denominator are integers.

Examples and identification

Common examples of irrational numbers include:

$\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ (square roots of non-perfect squares)
$\pi$ (pi)
$\frac{1 + \sqrt{5}}{2}$ (the golden ratio)

When written as decimals, irrational numbers continue forever without any repeating pattern.

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Identifying Rational vs Irrational Numbers

Here's a simple method to determine if a number is rational or irrational:

Write the number in decimal form
If the decimal terminates, it's rational
If the decimal continues forever but has a repeating pattern, it's rational
If the decimal continues forever with no repeating pattern, it's irrational

Converting decimals to fractions

Terminating decimals

Each digit after the decimal point represents a fraction with a denominator that's a power of 10.

For example: $10.589 = 10 + \frac{5}{10} + \frac{8}{100} + \frac{9}{1000} = \frac{10589}{1000}$

Recurring decimals

Converting recurring decimals requires more steps. Here's the systematic method:

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Worked Example 1: Converting $0.\overline{3}$ to a fraction

Step 1: Let $x = 0.3333...$

Step 2: Multiply both sides by 10: $10x = 3.3333...$

Step 3: Subtract the original equation: $10x - x = 3.3333... - 0.3333...$

Step 4: Simplify: $9x = 3$ , so $x = \frac{3}{9} = \frac{1}{3}$

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Worked Example 2: Converting $5.\overline{432}$ to a fraction

Step 1: Let $x = 5.432432432...$

Step 2: Since three digits repeat, multiply by 1000: $1000x = 5432.432432...$

Step 3: Subtract: $1000x - x = 5432.432... - 5.432...$

Step 4: Simplify: $999x = 5427$ , so $x = \frac{5427}{999} = \frac{201}{37}$

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General Rule for Recurring Decimals

The key is to match the number of repeating digits:

If one digit recurs, multiply by 10
If two digits recur, multiply by 100
If three digits recur, multiply by 1000
And so on...

Worked example: Identifying number types

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Worked Example: Classifying Numbers

Question: Which of the following are not rational numbers?

$\pi = 3.14159265358979323846264338327950288419716939937510...$
$1.4$
$1.618033989...$
$100$
$1.737373737...$
$0.02$

Solution:

Irrational - decimal does not terminate and has no repeated pattern
Rational - decimal terminates
Irrational - decimal does not terminate and has no repeated pattern
Rational - all integers are rational
Rational - decimal has repeated pattern (73)
Rational - decimal terminates

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Key Points to Remember:

The real number system is organised in a hierarchy: Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real, with Irrational numbers forming the other half of Real numbers
Rational numbers can be written as fractions $\frac{a}{b}$ where both $a$ and $b$ are integers and $b \neq 0$
Rational numbers as decimals either terminate or have a repeating pattern
Irrational numbers cannot be written as fractions and have non-terminating, non-repeating decimal representations
To convert recurring decimals to fractions, multiply by the appropriate power of 10 based on how many digits repeat, then subtract and solve

Real, Rational, and Irrational Numbers (Grade 10 NSC Matric Mathematics): Revision Notes